Prove/Find Counterexample: Intro to Set Theory

[mbcsantin]

Homework Statement

Prove or find counterexamples. For any sets A, B, C in a universe U:

if A union C contained B union C then A contained B

Homework Equations

none.

The Attempt at a Solution

im just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks

Let A be the empty set, and let B = C
Then A union C = B and
B union C = B so,
A union C contains B union C, but A does not contain B because A is the empty set and B is not.

 

[e(ho0n3]

Looks right to me. Just one small note: You should state that B = C is not empty at the beginning.

 

[mbcsantin]

Looks right to me. Just one small note: You should state that B = C is not empty at the beginning.

alright. thank you so much!

 

[mbcsantin]

Looks right to me. Just one small note: You should state that B = C is not empty at the beginning.

But what if I use the element proof for this..

Supposed that A is a subset of B.

Let x is an element of A u C.
therefore, x is an element of A and x is an element of C.
Since A is a subset of B by the definition of containment, x is an element of B.
Since x is an element of B and x is an element of C, we have x is an element of B u C. so any element of B u C is also in A u C. therefore, A u C is a subset of B u C.

Would this be right?

 

[HallsofIvy]

You are giving a counter example. You don't need a general proof, just any single counter example. You could just take A= {}, B= {1}, C= {1}.

 

[mutton]

Let x is an element of A u C.
therefore, x is an element of A and x is an element of C.

No, that implies x is an element of A or x is an element of C.

Anyway, what you did here is irrelevant to your original question.